Well-posedness of three-dimensional isentropic compressible Navier-Stokes equations with degenerate viscosities and far field vacuum

TL;DR

This paper establishes local well-posedness for 3D isentropic compressible Navier-Stokes equations with density-dependent viscosities and vacuum at infinity, addressing an open problem and contrasting with classical constant viscosity results.

Contribution

It identifies initial data conditions for local solutions with vacuum and finite energy, solving a previously open problem related to degenerate viscosities.

Findings

Local regular solutions exist with vacuum and finite energy.

Global solutions with decay of velocity are not possible under these conditions.

Addresses an open problem in degenerate viscous flow theory.

Abstract

In this paper, the Cauchy problem for the three-dimensional (3-D) isentropic compressible Navier-Stokes equations is considered. When viscosity coefficients are given as a constant multiple of the density's power ($\rho^\delta$ with $0<\delta<1$), based on some analysis of the nonlinear structure of this system, we identify the class of initial data admitting a local regular solution with far field vacuum and finite energy in some inhomogeneous Sobolev spaces by introducing some new variables and initial compatibility conditions, which solves an open problem of degenerate viscous flow partially mentioned by Bresh-Desjardins-Metivier [3], Jiu-Wang-Xin [11] and so on. Moreover, in contrast to the classical theory in the case of the constant viscosity, we show that one can not obtain any global regular solution whose $L^\infty$ norm of $u$ decays to zero as time $t$ goes to infinity.